


I hope you find find this page
interesting and informative. I will be adding to it from my notes
and future studies as time permits so please come back often. 
So far, I have been concentrating mainly on finding the basic solutions
for the different orders. There is much left to discover about the
characteristics of the individual orders. Share with me the excitement of
the search.
If you are also interested in Magic Stars, I would
like to hear from you.
March 1, 2005: NEW 4 pages added. See bottom
of contents!
April 4, 2007: NEW(page 9x5 stars) added.
May 18,2007: New section on early solvers added to
Order 6 Stars.
2010 Note: Because this section was a prominent part of the original
web site, I have chosen to include it on this renamed Magic
Hypercubes site. 
On this page
Introduction 
A basic definition of Magic Stars and the similarity
to Magic Squares. Includes a diagram of the 3 Order9 patterns and
shows the order that numbers are assigned to the lines.

Basic Equivalent Solutions 
An explanation of which solutions are considered
basic and which are equivalent solutions.
The two requirements for a basic solution and converting an
equivalent to a basic solution.

Complements Index Numbers 
Each solution has a complement. If the solution is
basic the complement is an equivalent and must be normalized to
arrive at it's basic solution.
Includes a diagram of four Order6 solutions to illustrate the
above.

References for Magic Stars 
Sixteen different diagrams from Order5 to Order11d
Also shown is the solution number and the total number of solutions.


Other pages in this section
1. A
Magic Star Definition 
What is a Magic Star?
Here is a formal definition and an explanation of terms used in my
discussion of magic stars.
Included also are comparisons between the different orders. 
2. Examples
of Magic Stars 
Sixteen different diagrams from Order5 to Order11d
Also shown is the solution number and the total number of solutions. 
3.
Examples of Magic Stars  2 
Sixteen different diagrams from Order12a to
Order14e.
Also shown is the solution number and the estimated total number of
solutions. 
4. Big Magic
Stars 
1 solution for pattern A of orders 15 to 20. Also
blank graphs of the other patterns for each order. 
5. Order5
Magic Stars 
Order5 is not a pure magic star but there are 12
solutions using numbers 1 to 12 but omitting numbers 7 and 11.
Another 12 solutions leave out the 2 and 6. 
6. Order6
Magic Stars 
A list of the 80 basic solutions along with
characteristics. 20 sets of 4. Supermagic stars. A tribute to H. E.
Dudeney. 
7.
Order6 Solution List 
A tabular listing of the 80 basic solutions along
with additional features. 
8.
H. E. Dudeney Features 
Features of the order6 magic stars that were first
reported by H. E. Dudeney in 1926. 
9. Order7
Magic Stars 
General characteristics. Lists of the 72 basic
solutions for each of the 2 patterns. 
10. Order8
Magic Stars 
General characteristics. Lists of the 112 basic
solutions for each of the 2 patterns. 
11. Order9
Magic Stars 
General characteristics. Condensed lists of basic
solutions for each of the 3 patterns. 
12. Order10
Magic Stars 
General characteristics. Condensed lists of basic
solutions for each of the 3 patterns. 
13. Order11
Magic Stars 
General characteristics. Condensed lists of basic
solutions for each of the 4 patterns. 
14. Prime
Magic Stars 
Magic Stars consisting of prime numbers. Lists of
minimal solutions consecutive primes solutions for orders 5 and 6. 
15. Prime
Magic Stars  2 
Diagrams and lists of minimal solutions
consecutive primes solutions for orders 7 A B and 8 A B. 
16. Unusual
Magic Stars 
Patterns with combinations of stars or more then 4
numbers per line. 
17. Isolike
Magic Stars 
Stars that are transformations of magic squares.
Also plusmagic and diammagic squares. 
18. Magic 9 x 5
Order6 Stars 
This new page (April 2007) shows 1 solution for each
magic constant (S) of 46 to 54 of the 5 numbers in each of 9 lines
magic hexagram. Note that most magic star pages on this site deal
with only 4 numbers/line. 
19. 3D Magic
Stars 
This magic 8point star contains 12 lines of 3
numbers, plus many other lines as a result of the missing numbers of
the series forming a nucleus and two satellites. 
20. Trenkler
Stars 
Marian Trenkler defines stars as of 2 types. He also
defines almostmagic weaklymagic stars. 
21. Magic
Star Puzzles 
Pictures showing star (and other magic object)
puzzles. Some quite old.
Also, some pencilandpaper puzzles of magic objects. 
22. Star
Updates 
This page, started in March, 2005, will contain
material added to this site or links to subpages of such material. 
a.
Simon WhiteChapel 
With emails starting in 2001, Simon presents
solutions for pattern A of magic stars from 15 to 100. 
b.
Jon Wharf 
With emails starting in 2003, Jon confirms the total
solution count for all orders and patterns from 6 to11, and provides
the total solution count all order 12 patterns. He also supplies
some solutions for all patterns of orders 13 and 14. 
c.
Andrew Howroyd 
First contacted me in February, 2005. He also
confirms all total solution counts, and investigated permutations
between patterns of orders 10 and 11. 
Introduction
Magic stars are similar to Magic Squares in many ways. The
order refers to the number of points in the pattern. A standard magic star
always contains 4 numbers in each line and in a pure magic star they
consist of the series from 1 to 2n where n is the order of the star.
The diagram above demonstrates also how the numbers are
assigned to the cells one line at a time.
Note also, all orders greater then six consist of multiple patterns, each
of which consist of a different list of basic solutions. I have found no
reference in the literature to this fact
Of course, some star patterns have more then two line
crossings (plus the two points) per line. See, for example, orders 9b and
9c above. In these cases, we could assign more then 4 numbers to a line in
such a way that all lines sum the same. These too would be magic stars.
However, to keep the variations to a manageable number, my studies have
been limited to the cases where only the perimeter line junctions (i.e.
the points and valleys) have numbers assigned to them.
Pattern naming convention. Originally I had rather
arbitrarily assigned names a, b, c, etc to the various patterns of an
order of magic star. In January, 2001, Aale de Winkel suggested a
systematical way of applying these labels.
Imagine the points of a star diagram as being points on a circle. Then
each point in turn is connected by a line to another point, by moving
around the circle clockwise. If we step once and connect to the second
point, the pattern is called 'A'. Stepping twice, and connecting to the
third point, produces pattern 'B'. etc.
Another way to look at this subject:
'A' has 4 intersections per line, 'B' has 6, 'C' has 8, 'D' has 10, and
'E' (required for orders 13 and 14) has 12 intersections per line.
By Feb. 16, 2001, all relevant pages have been revised to
show the new pattern names.
Basic
Equivalent Solutions
Each star has solutions that are apparently different but
in fact are only rotations and/or reflections of the basic solution. The
order10 star with its 10 degrees of rotational symmetry, each of which
may be reflected, has 20 apparently different solutions. Only one of these
is considered the basic solution.
Two characteristics determine the Basic Solution.
Any magic star solution may be converted to a basic
solution by normalizing it, i.e. performing the necessary
rotations and/or reflections so the solution confirms to the above
criteria.
Any magic star can be converted to another magic star by
adding or multiplying each number in the star by a constant. This feature
also applies to magic squares.
Of course, the resulting star would not be pure (normal)
because the number series would no longer be consecutive.
Complements
Index Numbers
Any magic star can be made into another magic star by
complementing each number of the original star in turn. This is done by
subtracting each number from n + 1. In the case of the order6
star, which uses the numbers 1 to 12, you subtract each number from 13 to
obtain the new number.
Diagram 
Solution # 
a 
b 
c 
d 
e 
f 
g 
h 
i 
j 
k 
l 
Compl.
Sol. # 
Compl.
Pair # 
description 
A. 
38 
1 
9 
11 
5 
4 
10 
7 
6 
12 
3 
8 
2 
79 
32 
How solutions are written 
B. 
39 
1 
9 
12 
4 
3 
11 
8 
7 
10 
5 
6 
2 
78 
33 
The next solution in index order 
C. 

12 
4 
1 
9 
10 
2 
5 
6 
3 
8 
7 
11 


Not a basic solution 
D. 
78 
5 
2 
10 
9 
1 
4 
12 
3 
6 
7 
8 
11 
39 
33 
Diagram c. normalized by rotation 2 positions
clockwise, then a horizontal reflection 
If the original is a basic solution, the complement star
will not be a basic solution. It is an equivalent, but after
normalizing, it will be another basic solution. When enumerating solutions
for magic squares, the complements are also counted as basic solutions. We
will follow the same convention when counting and indexing the magic star
solutions. This means that the number of solutions for each order of magic
star must always be an even number and the number of complement pairs is
exactly half the number of total solutions. To put it another way, all
basic solutions come in pairs which are complements of each other.
The fact that all solutions have a pair partner determine
some characteristics for a particular order. For example, if you find a
solution with all odd numbers at the points, you can be confident another
solution exists that has all even numbers at the points. Likewise, if a
solution exists that has all the low numbers at the points, another one
exists that has all the high numbers.
The complementing process works for all magic squares and
all magic stars even if the numbers are not consecutive or do not start at
1. In such cases, the complementary number is obtained by subtracting from
the sum of the first and last number in the series used. Even prime magic
stars have a compliment, although because compliments of many of the prime
numbers are not prime numbers, the resulting magic star will not be a
prime magic star.
Order5 magic stars come in pairs where the points of one
member appear as the valleys of the other member. I call these pairs Pcomp
because they are complements of each other, but not in the accepted sense.
References for
Magic Stars
Order6 is the smallest pure magic star and the only one
with only one star pattern (a fact not mentioned in the literature). In
fact, in contrast to the voluminous literature for magic squares spanning
100's of years, there has been very little published on magic stars. The
two main sources of information I have been able to locate are:

H.E.Dudeney, 536 Puzzles Curious Problems,
Scribner's 1967. Lots of info on order6.

Martin Gardner, Mathematical Recreations column of
Scientific American, Dec. 1965, reprinted with addendum in Martin
Gardner, Mathematical Carnival, Alfred A. Knoff, 1975. Mostly on
order 6, but mention made of total basic solutions for orders 7 8 (also
corrected number for order6).
Marián Trenkler of Safarik University, Kosice, Slovakia
published a paper on Magic Stars.
It is called Magicke hviezdy (Magic stars) and appeared in Obsory
matematiky, fyziky a informatiky, 51(1998), pages 17. (Obsory = horizons
(or line of sight) of mathematics, physics and informatics.
Magic squares, perhaps because they are quite ordered
structures, have been studied for centuries. In contrast, magic stars have
few similarities between orders, or for that matter even between patterns
within an order. This makes it necessary to study each pattern
individually
My studies (to March 1998) include all basic solutions for
orders 5 to 11 and most solutions for order12, a total of 20 patterns.
Also, many solutions for each of the 10 patterns of orders 13 and 14.
Here are 16 sample magic
stars for all orders and patterns from five to eleven and
here are the 14 patterns for orders twelve
to fourteen. And here are 6 examples of pattern A stars of orders 15 to
20.
Also, be sure to check out Definitions and
Details, and the Order6 page. Over time, I
intend to add more pages, covering details of the different orders, and
including lists of solutions. So please check this site periodically.
